Notes:
An algorithm for evaluation of friction in general contact-impact interfaces is described. The algorithm is based on an explicit finite-element method. Coulomb's friction law is assumed. The defence node algorithm is used such that the sticking condition can be imposed with the Lagrange multiplier method even in explicit dynamic analysis. The algorithm is supposed to be applicable in general situations, including large deformations of the contact-impact bodies and large relative sliding between the contact-impact boundaries. Numerical results are presented to demonstrate the performance of the algorithm.

Notes:
In the paper we have developed a new formulation for solution of structural-acoustic coupling problems by boundary elements using the multiple reciprocity method. It is assumed that the structure is composed of plate components and is excited by the external or the internal noise source. The efficiency of the proposed formulation becomes especially remarkable if the boundary-value problem is to be solved repeatedly for different values of frequency. The accuracy of the numerical computations has been compared with the analytical solution in a test example.

Notes:
A generalization of the r(1-m)/m strain singularity of higher-order isoparametric elements is presented. It is shown that, by variable placement of the side nodes between their original and singular positions, the point of singularity sensed by the element can be controlled. The transition elements have a strain singularity outside their domain. The singular and non-singular elements are elements are special cases of the general mapping. The transition elements, together with the singular isoparametric elements, can be used for solving crack problems.

Notes:
In the design of members with flaws, it is necessary to keep the stress intensity factor K of any sharp crack below the fracture toughness Kcr of the materials. Stress-intensity factor equations for the more common basic specimen geometries and various loading conditions are available in the literature. The application of these equations to complex structures involves geometric problems such as the identification of the outline of each member and the sizing of the equivalent specimen for each flaw. The paper gives a response to such difficulties.

Notes:
A simple, yet storage-effective ‘linear’ programming code is given. The assumption of non-negative variables is bypassed without increasing the size of the problem. Furthermore, the objective is allowed to be summed over not just linear, but also concave, functions. A specific truss topology optimization example is shown.

Notes:
We propose in this paper a nine-point, fourth-order difference method for the numerical solution of the quasilinear Poisson equation \documentclass{article}\pagestyle{empty}\begin{document}$$ Au_{zz} + \frac{1}{r}u_r + Bu_{rr} = f\left({r,z,u,u_r,u_z} \right) $$\end{document} with appropriate boundary conditions. The method is based on five evaluations of f. The numerical results of four problems obtained using this method are listed. The results demonstrate the fourth-order accuracy of the method.

Notes:
Singularly perturbed second-order elliptic equations with boundary layers are considered. These may be considered as model problems for the advection of some quantity such as heat or a pollutant in a flow field or as linear approximations to the Navier-Stokes equations for fluid flow. Numerical methods composed of central-difference operators on special piece-wise-uniform meshes are constructed for the above problems. Numerical results are obtained which show that these methods give approximate solutions with error estimates that are independent of the singular perturbation parameter. An open theoretical problem is posed.

Notes:
In the paper we present a superconvergent patch recovery technique for obtaining higher-order-accurate finite-element solutions and thus a postprocessed type of L2 norm error estimate. Two modifications make our procedure different from the one proposed by Zienkiewicz and Zhu (1992), in which higher-order-accurate derivatives of the finite-element solution at nodes are determined. Firstly, the recovery process is made for element, not for nodes. An ‘element patch’, which represents the union of an element under consideration and the surrounding elements, is introduced. Secondly, the local error estimate is calculated directly from the improved solution for this element. Numerical tests on both 1D and 2D model problems show that this method can provide an asymptotically exact a posteriori L2 norm error estimate if the used element possesses superconvergent points for the solutions.

Notes:
The paper presents a probabilistic boundary element method for analysis of structural responses to static loading, when the shape parameters of structures are considered as random variables. We can use this method to evaluate the mean values and standard deviations of the responses, and estimate the stochastic errors of structural character resulting from manufacturing errors.

Notes:
In the paper we address the problem of 2D adaptive quadrilateral mesh generation by using the variational principles. We first find the variational integral which generates the known grid system of curve-by-curve error equidistributions. We then use the same intergral to generate a new adaptive grid system which is superior to the known system in that the new system produces smoother meshes. Moreover, the new integral may be combined linearly with existing smoothness control integrals to yield more robust adaptive grid systems. Numerical results and comparisons are reported.

Notes:
The generalized integral transform technique is used to reduce eigenvalue problems described by partial differential equations to algebraic ones, that can be solved by existing codes for matrix eigensystem analysis. The method is illustrated for the operators that correspond to heat and mass diffusion, but can be employed in different fields. Three examples demonstrate the potential of the method.

Notes:
The paper presents a new method for the numerical solution of transient linear heat conduction problems. In the proposed method, the transient linear heat conduction equation is first integrated with respect to time over subsequent intervals. Then the resulting set of elliptic equations is discretized in space according to a finite-element procedure. The method is unconditionally stable. At the end of the paper, the effectiveness of the proposed method is assessed through some numerical examples.

Notes:
Use of upwind finite element methods was so far confined to forced convection problems. The present work is devoted to the application of false diffusion techniques to phenomena involving natural convection. A range of Rayleigh numbers in which the discretization by the conventional Galerkin scheme fails is considered. A false diffusion method similar to the Streamline upwind technique has been employed to avoid cross-wind diffusion. The upwind parameter is changed continually until a true solution is obtained. The present scheme provides convenient means for obtaining gradually improved guesses from which a more accurate solution can be obtained using the conventional Galerkin method. The criterion for the determination of a true solution is established, and hot-wall Nusselt number values are documented in tabular form.

Notes:
An interface finite element for non-linear analysis of frictionless contact problems is presented. A constitutive equation relates stresses in the interface layer to the deformation gradient with respect to an imaginary reference configuration where the layer thickness is constant and finite. The equation of equilibrium is written in the same configuration, while the boundary conditions involve stresses related to an actual reference configuration used in the formulation for the contacting bodies. Finite element discretization is introduced for the layer in order to calculate the unknown field of displacements. Computational examples demonstrate properties of the contact element and its range of applicability in the analysis of large deformation contact problems including relative sliding of curved surfaces.

Notes:
An approach is presented for the formulation of the unilateral contact constraint in the presence of contact surface discontinuities. Such discontinuities may be due to physical corners on the surfaces of contacting bodies, or may be introduced by a discretization process (e.g. finite elements). It is asserted that a strong analogy exists between this problem and the one describing inelastic evolution in the presence of a discontinuous yield surface. This analogy is exploited to produce an effective treatment of the frictionless corner problem, complete with an effective augmented Lagrangian implementation for accurate constraint enforcement.

Notes:
The paper presents boundary integral equation to two-dimensional elasticity with the stress component σijtitj as one of the boundary values, where ti are direction cosines of the tangent on the boundary. This form of BEM has an advantage in that the stress component σijtitj on the boundary can be calculated directly from the numerical solution. The present formulation for planar problems uses two kernels, one of which is logarithmic singular and the other is 1/r singular. The effectiveness of the approach is also discussed through some test examples.

Notes:
A finite difference scheme is derived to solve for the probability density that a moving point, whose velocity is continually pertubed by Gaussian white noise, reaches a given target within a given time period. The numerical scheme is applied to the problem of finding the probability that two moving spheres collide.

Notes:
Numerical viscosities of finite-difference schemes are usually obtained from truncation-error analyses based on Taylor series expansions. Here we observe that numerical viscosities can also be obtained very simply and directly from the growth factor ξ in a conventional Fourier stability analysis. A general formula is derived for the numerical viscosity in terms of the first and second derivatives of ξ with respect to the wavenumber k, evaluated at k = 0. A single Fourier analysis therefore suffices to determine both stability limits and numerical viscosities.

Notes:
Based on the concept of the Laplacian matrix of a graph, this paper presents the SGPD (spectral graph pseudoperipheral and pseudodiameter) algorithm for finding a pseudoperipheral vertex or the end-points of a pseudodiameter in a graph. This algorithm is compared with the ones by Grimes et al. (1990), George and Liu (1979), and Gibbs et al. (1976). Numerical results from a collection of benchmark test problems show the effectiveness of the proposed algorithm. Moreover, it is shown that this algorithm can be efficiently used in conjunction with heuristic algorithms for ordering sparse matrix equations. Such heuristic algorithms, of course, must be the ones which use the pseudoperipheral vertex or pseudodiameter concepts.

Notes:
A sixth-order and a ninth-order method are developed for the numerical solution of special non-linear third-order boundary value problems y‴ = ø(x,y), a〈 x 〈 b, y(a) = A1, y″(a) = B2 and y(b) = A2. The method arise from a four-point recurrence relation. Convergence analysis of the sixth-order method is discussed. The methods are tested on a problem. Modification to these methods are obtained for problems with boundary conditions of the form y(a) = A1, y'(a) = B1 and y(b) = A2.

Notes:
A finite element program is developed as a tool to analyse shells of revolution with local non-linearities. In reality, shells of revolution often exhibit local deviations, like a cut-out, a junction and/or an imperfection. The stress concentration around a local deviation may produce plasticity and/or geometric non-linearities in the surrounding region. The analytical model consists of three different types of elements: rotational, transitional and general. The rotational shell elements are used in the region where the shell is axisymmetrical and linear, while the two-dimensional general shell elements are deployed in the deviation region where non-linearities may occur. Transitional shell elements connect the two distinctively different types of elements to achieve displacement field continuities. The solution using the local-global system with appropriate condensation and a predicted stress incremental procedure is suggested. It is shown that the technique is a very attractive alternative to the entirely general element style analysis for axisymetric shell structures with local deviations.

Notes:
New developments of describing the theoretical basis towards an effective virtual-pulse (VIP) time integral methodology are proposed for general non-linear transient heat transfer problems. Primarily to validate the proposed methodology of computation, simple numerical test cases are provided and comparisons are also drawn with the implicit second-order-accurate Crank-Nicolson method. For the models tested, the proposed method has comparable or improved accuracy and stability characteristics. The VIP methodology introduced here for tackling non-linear thermal problems offers attractive features and is a viable alternative to traditional time-stepping practices. Efforts are underway to demonstrate the practical applicability to multi-dimensional thermal analysis.

Notes:
Finite element models based on discrete-layer theories are presented for the coupled-field analysis of laminated plates containing piezoelectric layers. The three displacements and the electrostatic potential are treated as unknowns in this formulation, which allows for piece-wise approximations of the variables through the thickness of each layer. Two specific models are demonstrated in which the transverse displacement is either variable or constant, and the in-plane displacements and potential take piece-wise linear approximations through the thickness. The models are applied to example problems with applied surface tractions and specified surface potentials. Good agreement is found with exact solutions.

Notes:
The paper describes an optimized computational implementation of a basic ‘building block’ for non-linear structural dynamic analysis programs: the combination of the modified Newton-Raphson iterative technique with an implicit time integration operator (in this case a member of the Newmark family), working in an incremental--iterative formulation for the equations of motion. The objective of this implementation is to attain improved computational efficiency, regarding both CPU time and memory requirements. The basic formulation and derivation are presented, along with the implementation details; the positive aspects related to the computational optimization are highlighted.

Notes:
On the basis of simple one-dimensional finite element analyses, comparison in simulations of uniaxial tension using explicit dynamic and implicit static tonnulations has been made. Results show that for materials having a Hollomon-type constitutive law with power-law strain rate sensitivity, the explicit dynamic method can be employed for the quasistatic simulations of tensile tests only below the critical test velocities or critical normalized material densities. These critical values are determined numerically as a function of material parameters.

Notes:
A new type of B3-spline interpolation is presented. This new formulation allows the user to introduce interior point and line supports in much the same way as in a conventional finite element formulation. The bandedness and efficiency of the standard B3-spline interpolation are retained. The new formulation can be incorporated into existing B3-spline formulations with very little effort. Several examples are presented to demonstrate the versatility of the new approach.

Notes:
A mixed variational principle is presented for the geometrically linear micropolar continuum. Then a suitable discretization and its implementation are discussed, resulting in an improved element behaviour for micropolar localization analysis. The intriguing element performance is demonstrated for the case of localization within a compression problem.

Notes:
A comparison is made of implementations of consistent tangent operators that arise in implicit integration of Von Mises and Drucker-Prager yield criteria. When computing the consistent tangent operator a matrix inversion has to be performed at integration point level. The consequences of different formulations of the consistent tangent operator on the numerical accuracy are assessed.

Notes:
An essential feature of the transient dynamic kernels for any time domain boundary integral equation is that they decay towards the static kernels for large time increments. The purpose of the paper is to show that the kernels of Mansur and Brebbia (1982), which are expressed in the form of generalized mathematical functions, do exhibit the correct decay property in contradiction to the claims of Israil and Banerjee (1990). Mansur and Brebbia express the fundamental solution wave discontinuity in terms of the generalized functions H(t) and δ(t), the Heaviside and Dirac delta functions, but do not explicitly demonstrate the required long time behaviour. In the 1990 paper the temporal discontinuity of the kernels is expressed by conventional functions in two time regimes and is shown to possess the correct long time behaviour, while it is claimed that the kernels in the 1982 paper do not. It is shown here that, assuming either constant or linear variation in time, usage of the generalized functions in the fundamental solution does yield the static fundamental solution for large time steps, as expected.

Notes:
Timoshenko beam elements have been the subject of numerous publications. The difficulty was that of arriving at a superconvergent element with four degrees of freedom, as is the case for the Bernuli-Euler classical beam element. Two different approaches are presented here for the derivation of the shape functions. The first is based on the flexibility matrix, where utilizing the unit load method, including the term that accounts for the shear deformations in the virtual work expression, the stiffness matrix is derived. Then, a second method is presented to derive the exact shape functions, directly from the differential equations of the Timoshenko beam theory. The resulting shape functions are the same in both methods.

Notes:
The convergence of stress maxima, computed directly from finite element solutions, is investigated with respect to a family of exact solutions characterized by varying degrees of smoothness. The performances of h- and p-extensions and the product and trunk spaces are evaluated and documented with respect to a family of benchmark problems. In uniform p-extensions a characteristic pattern in the convergence of stress maxima was observed. There does not appear to be a clear-cut advantage of the product space over the trunk space in this respect. The much faster convergence of stress maxima in the case of p-extensions, as compared with h-extensions, is evident from the results.

Notes:
Factorization procedures for the efficient solution of large sparse linear finite difference systems have been introduced recently. In these procedures the large sparse symmetric coefficient matrix of a certain structure is factorized exactly, yielding a direct solution method. Furthermore, approximate factorization procedures yeild implicit preconditioning iterative methods for the finite difference solution. The numerical implementation of these algorithms is presented and Fortran subroutines for the efficient solution of the resulting sparse symmetric linear system of algebraic equations are given.

Notes:
It is well known that the finite element discretization with a distributed mass matrix gives overstimates of the natural frequencies of a system. The note shows that these do not necessarily become progressively worse as the frequency increases.

Notes:
A technique is presented whereby numerical calculations of vibration modes can be improved. The paper looks at the classical two-dimensional wave equation using finite difference approximations. Analysis of the numerical dispersion of the approximations is used to develop a correction method. In general the numerical dispersion is dependent upon both the frequency and the direction of a wave, but if a 9-point formula is used the directional dependence is much reduced. This enables correction factors to be obtained using only the frequency of a vibration mode. The method was tested on the vibration of a square membrane and of an L-shaped region; in both cases a marked improvement in accuracy was obtained, at very little computational cost.

Notes:
This is a study about one of the core questions in the GMRES(k) method regarding the obtaining of vector yk for the least-squares problem, argminy |Hky - β(n)e1|2 (see Saad and Schultz1). We propose a simple but efficient approach to the resolution of this problem and a low cost computation of the residual and the residual norm, including both in a complete and detailed FGMRES(k) algorithm. The whole algorithm of minimization only involves two backward substitutions with triangular matrices and a dot product. The residual and the residual norm are computed, making use of results in the least-squares problem.

Notes:
Many processes in the sciences and in engineering are modelled by dynamical systems and - in discretized version - by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.

Notes:
When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.

Notes:
Sufficient conditions are obtained for the convergence of difference schemes for the numerical solution of the Cauchy problem for a heat conduction equation in two space variables. The sufficient conditions are derived in a form similar to those for the convergence of a sequence of linear positive operators in the Korovkin theorem. As an application it is shown that difference schemes that are widely used in practice can easily be checked for convergence by these conditions.

Notes:
A new approach to developing serendipity quadrilateral infinite elements is presented. Using these elements universal matrices for quasiharmonic equation are developed. For a particular member of the family these matrices are independent of the size and shape of the element. Using these matrices the element stiffness matrix can be generated in a simpler manner by taking into account the size and shape of the element.

Notes:
A new global secant relaxation (GSR)-method-based improvement procedure is used to improve the overall convergency performance of the modified Newton-Raphson iteration in carrying out the solution of discrete systems resulting from the finite-element discretization of a certain class of structural problems involving non-linear deformation behaviour.

Notes:
Theoretical and experimental analysis of free-surface electrohydrodynamic flow is fragmented and incomplete. Simulation studies of this phenomenon are further limited by the inherent complexities in the modelling process. In this note a mathematical model is developed to analyse free-surface electrohydrodynamic flow in two dimensions, and preliminary results of the simulation are described. The configurations examined include electrified conducting surfaces, the dielectrophoretic forces, and a conducting jet. The simulation is compared with analytical results in the first two investigations and is shown to be quite accurate. In the last simulation it is demonstrated that in the initial formation of a conducting jet, a 10 per cent increase in applied voltage results in about a 10 per cent increase in fluid velocity.

Notes:
The dual reciprocity boundary element method, first proposed by Nardini and Brebbia (1982, 1985), is a powerful technique for solving elliptic partial differential equations. Adopting this approach, a singular volume integral, which needs to be evaluated with a traditional boundary element method, can be converted into a boundary integral. However, when the governing equation is of a certain type, this conversion fails due to the singularities being introduced inside the physical domain and on the boundary arising by differentiating distance functions. We avoid these artificially created singularities by constructing a transformation which leads to improved numerical results.

Notes:
The paper demonstrates an approach to generate three-dimensional boundary-fitted computational meshes efficiently. One basic idea underlying the present study is that often similar geometries have to be meshed, and therefore an efficient mesh-adaption method, which allows adaptation of the topological mesh to the specific geometry, would be more efficient than generating all new meshes. On the other hand the mesh generation for Cartesian topologies has been shown to be a very simple task. It can be executed by connecting and removing brick elements to a basic cube. In connection with a so-called ‘Macro Command Language’, a high degree of automation can be reached when adapting topologically defined meshes to a surface. Furthermore, a high mesh quality has proved to be the key to good simulation results. During the mesh generation it is important to provide the possibility of modifiying the mesh quality and also the mesh density at any time of the meshing process. Using this generation method the meshing time is reduced - e.g. a computational mesh for a two-valve cylinder head can be generated within a few hours.

Notes:
A numerical problem which features prominently in the implementation of the boundary-element method (BEM) for the solution of heat-conduction problems arises from double-valued heat fluxes at boundary corners. This problem is readily dealt with by explicitly treating nodal heat fluxes as double-valued prior to assembly of the nodal equations. This formulation is very flexible as it also permits imposing discontinuous thermal conditions at any boundary node. However, at corner nodes where temperature only is prescribed, the scheme leads to two unknowns, while only one nodal BEM equation is available there. An additional equation which closely follows the previous work of Walker and Fenner is derived for these nodes, and this provides sufficient information to resolve the upstream and downstream values of the heat flux at the temperature node. The additional equation is general, is free of heuristic constraints, and is applicable through the wide range of acute, shallow, obtuse and re-entrant angles encountered in practice. Numerical examples are used to compare the present method with the Walker and Fenner approach and with analytical solutions. Results indicate both improved accuracy and generality, thus validating the present method.

Notes:
In this paper we present numerical experiments made to investigate the behaviour of the Newmark time-stepping scheme applied to non-linear dynamic systems. Our attention is focused on the instability and chaos in the Newmark scheme when it is applied to the equation ü + P(u) = 0, representing a non-linear elastic spring. Some unusual modes of behaviour, which are of substantial interest, have been observed. In the first case, a stable but chaotic solution is found. In the second case, while a stable solution is obtained with a certain time step, an unstable solution is found by decreasing the time step. In the third case, instability is triggered by neglecting the initial acceleration. A simple modification of the Newmark scheme is proposed which keeps the energy constant for the equation ü + P(u) = 0 and thereby guarantees unconditional stability. Numerical examples in support of such an energy-conserving scheme are presented.

Notes:
An adaptive refinement/derefinement algorithm of nested meshes is presented. Some definitions are introduced. The main properties of the derefinement algorithm are remarked upon and its efficiency is shown through two numerical examples: a time-dependent convection-diffusion problems with dominant convection and a quasistationary problem.

Notes:
Fracture in heterogeous solids can be simulated with finite elements. We compare simulations of mechanical breakdown using meshes based on two types of geometrical elements: quadrilaterals and hexagons. The different co-ordination of the meshes leads to important differences in the load-carrying geometry close to failure. Hexagonal elements give a better representation of failure in brittle solids, whereas quadrilateral elements have a resemblance to fibrous solids.

Notes:
A note on the effects of a weak discontinuity in the forcing function g(x) of a singular, integral equation of the first kind and the resulting strong discontinuity that can appear in the solution f(x) is presented.

Notes:
The paper deals with a B-type strain projection method with specific reference to axisymmetric incompressible solids. Essentially the \documentclass{article}\pagestyle{empty}\begin{document}$ \overline{\overline {\rm{B}}} $\end{document} (B double bar) method, which is reported in the present study, is based on Hughes' B-bar approach utilizing modified shape functions to represent the radial displacement and geometry. As a consequence of adopting these procedures the \documentclass{article}\pagestyle{empty}\begin{document}$ \overline{\overline {\rm{B}}} $\end{document} method results in an exact satisfaction of incompressibility constraints. The rationale of the \documentclass{article}\pagestyle{empty}\begin{document}$ \overline{\overline {\rm{B}}} $\end{document} method leads to the modified mean-dilatation approach for the case of an axisymmetric incompressible problem. It is interesting to note in the present work that the explicit influence of enhanced hoop strain is eliminated from the discrete divergence condition.